PHYSICS OF PLASMAS 14, 092110 共2007兲

Gyrokinetic simulations of collisionless magnetic reconnection
B. N. Rogersa兲 and S. Kobayashi
Department of Physics and Astronomy, Dartmouth College, Hanover, New Hampshire 03755, USA

P. Ricci
Centre de Recherches en Physique des Plasmas, École Polytechnique Fédérale de Lausanne,
Association EURATOM-Confédération Suisse, CH-1015 Lausanne, Switzerland

W. Dorland, J. Drake, and T. Tatsuno
University of Maryland, College Park, Maryland 20742, USA

共Received 3 July 2007; accepted 27 July 2007; published online 20 September 2007兲
Linear and nonlinear gyrokinetic simulations of collisionless magnetic reconnection in the presenceof a strong guide field are presented. A periodic slab system is considered with a sinusoidally
varying reconnecting magnetic field component. The linear growth rates of the tearing mode in boththe large and small ⌬⬘ regimes are compared to kinetic and fluid theory calculations. In the
nonlinear regime, focusing on the limit of large ⌬⬘, the nonlinear reconnection rates in the
gyrokinetic simulations are found to be comparable to those obtained from a two-fluid model. In
contrast to the fluid system, however, for Ti Ⰷ Te and very small initial perturbation amplitudes, thereconnection in the gyrokinetic system saturates in the early nonlinear phase. This saturation can beovercome if the simulation is seeded initially with sufficient random noise. © 2007 American
Institute of Physics. 关DOI: 10.1063/1.2774003兴
I. INTRODUCTION

We present linear and nonlinear gyrokinetic simulationsof collisionless magnetic reconnection in the presence of astrong guide field. Our simulations are based on the GS2
code,1,2 which time evolves the nonlinear electromagnetic
gyrokinetic equations3 for both electrons and ions. Theseequations, together with Maxwell’s equations, govern theevolution of the electron and ion distribution functions 共depending here on 2 spatial and 2 velocity-space coordinates兲
and the self-consistent electric and magnetic fields. A summary of the full system of equations is given in Appendix A.
The GS2 formalism is based on the reduced magnetohydrodynamic 共MHD兲 ordering 共see, e.g., Ref. 4兲, in which thedeviations of the absolute levels of the density, temperatures,and the magnetic field 共for example兲 are assumed to be small
over the region of interest, though deviations in the gradientsof these quantities can be comparable to or larger than theequilibrium values. Such an ordering is appropriate to thestudy of strong guide field reconnection, in which the component of the magnetic field that reconnects is small compared to the total magnetic field strength. The magnetic fieldand plasma profile variations associated with reconnection inthis case, even in the nonlinear phase, are small compared tothe unperturbed background values. Similarly, the time variations associated with reconnection in the large guide-fieldregime are slow compared to the ion gyroperiod. These properties are both necessary for the gyrokinetic approach, inwhich one of the three velocity-space coordinates 共present ina full Vlasov treatment, for example兲 is eliminated from thesystem by a gyroangle average. This is computationally an
advantage relative to full-particle simulations because ita兲Electronic mail: barrett.rogers@dartmouth.edu

1070-664X/2007/14共9兲/092110/10/$23.00

avoids the necessity to time-resolve the gyroperiod of theparticles, which in the strong guide field limit is much fasterthan the time scale on which the modes of interest evolve. A
similar computational gain is made in GS2 by enforcing totalpressure balance, which eliminates compressional magnetosonic waves from the system while preserving the shearAlfvén dynamics that are essential to reconnection.
We consider a collisionless periodic slab geometry witha sinusoidally varying reconnecting magnetic field component By共x兲 ⬀ sin共kxx兲. Depending on the dominant wavelength of the tearing mode under consideration 共i.e., ky= 2␲ / Ly, where Ly is the length of the system in they-direction兲, this system allows us to explore reconnection ineither the limits of large ⌬⬘ 共relevant to the m = 1 mode intokamaks, for example, or the GEM Reconnection Challenge
Project5兲 or small ⌬⬘ 共relevant to tokamak tearing modes
with m ⬎ 1兲. We focus on a parameter regime of interest to
magnetic confinement fusion experiments in which ␳s ⬎ de,where ␳s = cs / ⍀ci is the ion sound Larmor radius, de = c / ␻ peis the collisionless electron skin depth, ␻2pe = 4␲ne2 / me,
cs2 = 共Ti + Te兲 / mi, ⍀ci = eB / 共mic兲. Both the linear and nonlinear regimes are considered. In the linear case, we compare
the gyrokinetic growth rates to analytical and numerical calculations based on kinetic and two-fluid models. Althoughreasonably good agreement between the two is found insome cases 共small Ti / Te and low ␤兲, some of the parameters
explored here 共Ti ⲏ Te and ␤ ⬃ 1兲 fall in a regime in which
existing analytic calculations are not clearly reliable. A rigorous treatment of these cases would seem to require the fullgyrokinetic model and include, among other things, bothelectron and ion finite Larmor radius 共FLR兲 effects. Althoughsuch an analysis is beyond the scope of the present article,
derivations of some of the analytic results to which we com-

14, 092110-1© 2007 American Institute of Physics

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092110-2

Phys. Plasmas 14, 092110 共2007兲

Rogers et al.

pare are given in Appendix B, and the problems with thesecalculations are discussed.
In the nonlinear regime, we focus on a large-⌬⬘ case that
corresponds approximately to the fastest-growing linear
mode evolved into the nonlinear regime. We find the reconnection rates in the gyrokinetic system are comparable tothose obtained from simulations based on a two-fluid model.
An exception arises for Ti Ⰷ Te and very small initial perturbation amplitudes, in which the nonlinear growth of themode in the gyrokinetic system saturates in the early nonlinear phase 共island widths comparable to ␳i兲. This saturation
can be overcome if small-amplitude random noise is added
to the initial configuration.
The results discussed in this paper build on a number ofother gyrokinetic studies of linear and nonlinear reconnection. Reference 6 applies gyrokinetic particle simulations tothe study small-scale magnetic islands 共widths comparable tothe electron skin depth de = c / ␻ pe兲 in the presence of a narrow 共about 10de兲 current sheet. References 7 and 8 explore
reconnection in the small ⌬⬘ case with gyrokinetic ions anddrift-kinetic electrons. In larger box sizes 共⬃64␳i兲 an instability with odd parity was found to dominate the system, incontrast to the expected even parity of the tearing mode. In
the large ⌬⬘ nonlinear simulations described here, such oddparity modes have not been observed.
This paper is organized as follows: In Sec. II we describe
the initial equilibrium and numerical simulation model. In
Sec. III we compare the GS2 simulation results to linear
theory calculations. We turn to the nonlinear regime in Sec.
IV, in which we address the reconnection rate and saturation
behavior. Our main conclusions are summarized in Sec. V.
The gyrokinetic equations used in the simulations are described in Appendix A, and the derivations of various linear
theory results are given in Appendix B.

The initial, unperturbed equilibrium state we consider isgiven by
By0共x兲 = By0 max sin共kxx兲,Bz0 Ⰷ By0 ,共1兲

n = n0 = const,

Ti = Ti0 = const,

Te = Te0 = const. 共2兲

For most of the numerical simulations discussed in this work
we take kx␳se = 0.2 共equivalent to a box size of Lx = 2␲ / kx
= 10␲␳se兲, where冑T e m ic=
mi eBz0

␳se =

cse
=
⍀ci

d2i =

c2
c 2m i=,␻2pi 4␲n0e2

␤e =8 ␲ n 0T e
2
Bz0

冑

␤e
di ,
2

.

Other definitions that will be used later include

␤=

d2e =

8␲n0共Ti + Te兲
2
Bz0

␤ 2
d ,
2 i
共4兲

,

c2 me 2= di .
␻2pe miNote that ␳se is the ion sound Larmor radius based on Te
while ␳s is based on Te + Ti.
A summary of the gyrokinetic and Maxwell equations
solved by the GS2 code is given in Appendix A. The temporal
discretization in GS2 to solve the linear terms in these equations is implicit and second order accurate in time. Nonlinearities are evaluated using an explicit Adams-Bashforthtime-stepping scheme. In the x and y directions 共the 2D planeof reconnection simulated here兲 a dealiased pseudospectral
algorithm is used. In the x-direction the simulations include
nx Fourrier modes with 64艋 nx 艋 8192, depending on parameters. In the y direction the nonlinear runs have a box size
Ly = 2.5Lx = 25␲␳se 共corresponding to ky,min␳se = 0.08兲 and 64
艋 ny 艋 256. The grid points in velocity space are chosen using Gaussian integration rules. Typical runs include 16 grid
points in the particle 共ion or electron兲 energies ⑀␣ = m␣v␣2 / 2
in the velocity range 0 艋 v␣ 艋 6vth␣, and 10 points in␰␣ = v储␣ / v␣. The simulations within this range appear to bewell converged and show little dependence on the velocityspace resolution. As discussed later, to facilitate parameterscans using modest numerical resources, an artificial electron
to ion mass ratio is used, either me / mi = 1 / 25 or me / mi
= 1 / 100.
III. LINEAR REGIME

The simulations described in this section address two
main cases

II. INITIAL EQUILIBRIUM AND SIMULATION MODELជ = B ẑ + B 共x兲ŷ,
B
z0
y0

2
␳s2 = 共1 + Ti/Te兲␳se=

共3兲Case I: kx␳se = 0.2,

mi/me = 100,␤e = 0.2,

Ti/Te = 共2.0 ⫻ 10−4,5兲,Case II: kx␳se = 0.2,

mi/me = 25,

Ti/Te = 共2.0 ⫻ 10−5,5兲.

共5兲␤e = 0.3,
共6兲

The simulation results in cases I and II are shown in Figs. 1
and 2, respectively. The black “⫹” symbols denote the GS2
values for very small Ti / Te Ⰶ 1 and the blue triangles represent Ti / Te = 5. Simulations at Ti / Te = 1 共not shown兲 are intermediate between the two, indicating the gyrokinetic growth
rates 共though not the eigenmode structure, as we show later兲are only very weakly dependent on Ti / Te. Since ␤e is held
fixed in both cases, this is equivalent to an insensitivity toeither the ion ␤i or the total plasma ␤, which for the parameters of case II, for example, varies from ␤ = 0.3 for Ti / Te
⯝ 0 to ␤ = 1.8 for Ti / Te = 5. Both cases I and II satisfy theinequality ␳s2 Ⰷ d2e 共equivalent to ␤ Ⰷ 2me / mi兲, which, aside
from being relevant to fusion devices, is required to allow
comparison to some analytic calculations discussed below.

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092110-3

Phys. Plasmas 14, 092110 共2007兲

Gyrokinetic simulations of collisionless magnetic reconnection

FIG. 1. 共Color online兲 Linear growth rates ␥ / ␥A vs ky␳se for case I with ␥A
given by Eq. 共10兲 and ␳se = cse / ⍀ci. The “⫹” are GS2 values for Ti / Te Ⰶ 1,
triangles Ti / Te = 5. The left and right dashed lines are Eqs. 共10兲 and 共11兲,
respectively, with Ti = 0. The solid lines are the numerical solution of areduced four-field fluid model for Ti = 0 共lower兲 and Ti = 5Te 共upper兲.

These analytic results are shown in Figs. 1 and 2 as thedashed and solid lines.
Numerical convergence tests have been carried out
across the full range of ky. The spatial resolution requirements in the x-direction for convergence of the growth rates
in the smaller-ky regime are modest 共nx ⬃ 64 for ky␳se = 0.08
and mi / me = 25兲 but increase strongly as ky is increased
共nx ⬃ 8192 for ky␳se = 0.16 and mi / me = 100兲. Such fine resolution levels in turn lead to severe time-step restrictions.
Consistent with the observed trend in the resolution, the linear theory calculations discussed later suggest the tearing
layer becomes narrower as ky is increased. The number ofmodes required at higher ky exceeds what one might expectfrom simple estimates, however, and is still under study.
The ky dependence of ␥ seen in Figs. 1 and 2 is governedby the tearing mode stability parameter ⌬⬘.9 This parameter
may be defined in terms of B̃x, the x-component of the magnetic field perturbation in the ideal regions outside the

FIG. 3. ⌬⬘ / kx given by Eq. 共8兲 vs ky / kx.

tearing layer. In our system, considering the layer at x = 0
共see, e.g., Ref. 10兲,B̃x ⬀ cos关D共kx兩x兩 − ␲/2兲兴sin共ky y兲,

共7兲
where ky is the wavenumber of the linear mode of interest,
and

⌬⬘ =

冋 册
⳵xB̃x
B̃x

x=0+

= 2kxD tan共␲D/2兲.

共8兲

x=0−

A plot of ⌬⬘ / kx is shown in Fig. 3 as a function of ky / kx
共solid curve兲. Consistent with Figs. 1 and 2 tearing modes
are unstable in the regime ky ⬍ kx 共here, ky␳se ⬍ 0.2兲 in which⌬⬘ ⬎ 0. For ky Ⰶ kx, ⌬⬘ / kx ⯝ 8k2x / 共␲k2y 兲 Ⰷ 1 共dashed curve兲.
The small-⌬⬘ regime lies to the right of the peak ␥ values in
Figs. 1 and 2; in this regime ⌬⬘ is sufficiently small so thatthe constant-␺ approximation is valid across the tearing
layer. The large-⌬⬘ regime includes the peak growth rates aswell as the regions to the left of the peaks in which thegrowth rates increase 共at these parameters兲 linearly with ky.
The ky value of the peak growth rate and the corresponding
value of ⌬⬘ can be estimated, for example, by balancing theexpression for ␥ that applies to the left of the peak 关in thelarge-⌬⬘ regime; see Eq. 共10兲兴 with the expression for ␥ thatapplies to the right of the peak 关in the small-⌬⬘ regime; see
Eq. 共11兲兴. This yields⬘ ⯝ 21/3␲2/3␳s−1/3d−2/3
.
⌬max
e

FIG. 2. 共Color online兲 Linear growth rates for case II with the same notation
as Fig. 1.

D ⬅ 冑1 − 共ky/kx兲2

共9兲

For the parameters of cases I and II this reduces to⌬max⬘ ⯝ 28 and ⌬max
⬘ ⯝ 20, respectively, or 共with Fig. 3 andkx␳se = 0.2兲, ky␳se ⯝ 0.06 for either case, consistent with thelocation of the peaks in Figs. 1 and 2.

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092110-4

Phys. Plasmas 14, 092110 共2007兲

Rogers et al.

A. Large ⌬⬘In the regime ␳s ⬎ de under consideration, various analytic results have been obtained in the limits of both large
and small ⌬⬘. At large ⌬⬘, the analytic results typically apply
in the regions to the left of the peak growth rates in Figs. 1
and 2 where ␥ ⬀ ky. The expression for ␥ obtained in Ref. 10,
for example, is given by␥=冉冊

2 1/3 1/3 2/3
k yde ␳s ␥A ,
␲

␥A =

冋 册

共10兲

kxBy0 maxdBy0=
冑4␲n0mi dx x=0 冑4␲n0mi .
1This result was obtained in Ref. 10 using an isothermal fluid
electron model and an ion model that includes leading-order
FLR effects for arbitrary Ti / Te but ignores sound-wave coupling under the assumption ␥ Ⰷ k储cs. 关A similar result,smaller than Eq. 共10兲 by a factor of 冑2 / ␲1/6 ⯝ 1.17, was also
obtained in Ref. 11 for Ti Ⰶ Te.兴 For convenience, a derivation of Eq. 共10兲 is given in Appendix B, where we also showthat a drift-kinetic electron treatment 共along with a Padé approximation to the electron plasma dispersion function兲 gives
the same result as the isothermal fluid electron model used inRef. 10. In the case of Ti Ⰶ Te Eq. 共10兲, shown as dashedlines on the left-hand sides of Figs. 1 and 2, agrees reasonably well with the gyrokinetic results. As we discuss in moredetail later, for small ky Eq. 共10兲 is also approximately
equivalent to an isothermal two-fluid model result shown as
the upper 共Ti / Te = 5兲 and lower 共Ti / Te = 0兲 solid curves.
These curves were obtained by solving numerically a set offour coupled reduced fluid equations described in Ref. 12,
which go beyond Eq. 共10兲 to allow for ion sound wave coupling and arbitrary ␤, but are formally valid only forTi Ⰶ Te. These curves are also essentially indistinguishable
from a numerical solution 共not shown兲 of the full, nonreduced two-fluid model that is discussed in Sec. IV. To generate the upper 共Ti / Te = 5兲 solid curves in the figures, wehave incorporated finite Ti effects into the fluid model of Ref.
12 by the replacement Te → Te + Ti. This is equivalent to assuming isothermal ion and electron equations of state and isconsistent with the Ti / Te dependencies of Eqs. 共10兲 and 共11兲.
It is not, however, consistent with the gyrokinetic growth
rates, which for our parameters show a much weaker dependence on Ti / Te. Indeed, at large ⌬⬘ the analytic and fluid
model growth rates increase with Ti / Te while the gyrokinetic
results display the opposite trend.
B. Small ⌬⬘Turning now to the case of small ⌬⬘, the dashed lines on
the right-hand sides of Figs. 1 and 2 show the following
asymptotic result of Refs. 10 and 13 evaluated at Ti = 0:␥=1k y ␳ s␥ Ad e⌬ ⬘ .
␲

共11兲

Like Eq. 共10兲, for convenience this expression is derived inAppendix B from a kinetic ion treatment and either an isothermal drift-kinetic or fluid electron model. A similar result

was also obtained in Refs. 14 and 15, aside from the replacement ␳s → ␳se and an overall factor of 冑␲ / 2 ⯝ 1.13. The GS2
results indicate that ␥ increases much more weakly with
Ti / Te than either Eq. 共11兲 or the two-fluid model. We notethat earlier versions of the two fluid model16,17 that neglected
the electron gyroviscous terms in the electron momentum
equation show much poorer agreement with the gyrokinetic
simulations in the collisionless small-⌬⬘ limit.
C. Limitations of the analytic and fluid model results

The analytic and fluid model results shown in Figs. 1
and 2 rely on some simplifying assumptions that are notclearly valid in our parameter regime. As we now explain, asimple isothermal or adiabatic equation of state cannot bestrictly correct in both the innermost electron layer where␥ ⲏ k储vthe and the outermost zone in which ␥ Ⰶ k储vthe 关notek储 = kyBy共x兲 / B ⯝ k⬘储 x is x-dependent within the tearing layer;
see Eq. 共B3兲兴. Similarly, the nongyrokinetic calculations neglect electron FLR effects, the electron polarization drift, andion sound wave coupling 共the fluid model includes the latterbut only in the zero-ion-FLR limit兲; all effects that, forTi ⬃ Te, become non-negligible at essentially the same point.
Since ion FLR and finite-␤ effects at Ti = 5Te are also nonnegligible, reliable analytic calculations in our higher-Ti parameter regime are thus a formidable challenge and have yet
to be carried out.
Electron FLR effects are potentially important because
the electron motion 关see, e.g., the expression for the current
density given by Eq. 共B15兲兴, for either large or small ⌬⬘,varies on the scale length

␦=

␥
␥de
=
,
k⬘储 vth ky␳s␥A

vth =

冑

Te + Ti.
me

共12兲The neglect of electron FLR effects requires that this scale,
which at large ⌬⬘ is a weakly decreasing function of Ti / Te,
exceed ␳e; a condition that for the parameters of cases I andII and large ⌬⬘ is marginally satisfied at Ti = 0 but is violated
at Ti / Te = 5. Similarly, at Ti / Te ⬎ 1, E储 and ␾ also exhibits
fine-scale structure for x ⬃ ␦ 关see Eq. 共B21兲兴:E储 ⯝ E储共0兲

冢

冣

Te 2
x
Te + Ti.
␦2 + x2

␦2 +

共13兲

This expression, similar to the behavior noted in Ref. 18,
predicts that E储 for Ti / Te ⬎ 1 drops by a factor of Te / 共Te+ Ti兲 on the ␦ scale. For larger x Ⰷ ␦ 关see Eq. 共B22兲兴 E储 thendecays to zero on the ␳s scale. The consistency of this withthe simulations can be seen in Fig. 4, which shows E储 / E储共0兲
in the linear regime for the case II parameters at ky␳se= 0.05 共large ⌬⬘兲. The width of the narrow peak at x = 0 for
Ti / Te = 5 共solid line兲 is comparable to ␳e and is broader than␦, consistent with the expectation that ␳e effects are becoming important. At Ti Ⰶ Te 共dashed line兲, such fine-scale structure is absent as expected from Eq. 共13兲.
A related effect, which only becomes important at finite
Ti / Te, is the electron polarization drift 关see Eqs. 共B30兲 or
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092110-5

Gyrokinetic simulations of collisionless magnetic reconnection

FIG. 4. E储 / E储共0兲 for case II with ky␳se = 0.05 and Ti / Te = 5 共solid兲, Ti / Te
= 2 ⫻ 10−5 共dashed兲. Sub-␳s structure appears only for Ti / Te ⬃ 1 or larger.

ñe,pol ⬃ −n 0e 2 2␳ ⵜ ␾.
Te e

共14兲

When k⬜␳i Ⰶ 1 and the ions are magnetized, the contributionof ñe,pol is negligible compared to the ion polarization drift.
At finite Ti, however, the ion response for x ⬃ ␦ Ⰶ ␳i becomes
adiabatic 关see Eq. 共B1兲, neglecting ⌫0兴,ñi ⬃ −

n 0e
␾.
Ti

共15兲

Estimating ⵜ2␾ ⬃ ␾ / ␦2, the contribution of Eq. 共14兲 is foundto be negligible compared to Eq. 共15兲 when 共Ti / Te兲␳2e Ⰶ ␦2.
This condition may be combined with the electron FLR restriction ␳2e Ⰶ ␦2 to yield
共1 + Ti/Te兲␳2e␦2=

␤ k2y ␳s2␥A2
Ⰶ 1.
2 ␥2

共16兲

Again, at large ⌬⬘, this condition for the case I and II parameters is marginally satisfied at Ti Ⰶ Te but is violated forTi / Te = 5. With the expression for ␥ given by Eq. 共10兲, it may
also be written as ␤ / 2 ⬍ 共me / mi兲1/4, a condition that was alsoemphasized in Refs. 16 and 17. Due to the k2y factor in the
numerator, Eq. 共16兲 is even more difficult to satisfy at small⌬⬘ 共large ky兲 than at large ⌬⬘ 共small ky兲. Thus, even for Ti
Ⰶ Te, we would expect electron FLR effects—absent in Eq.
共11兲 and the two-fluid model—to play an even greater role.
Two other effects absent from Eqs. 共10兲 and 共11兲,
共though not the two-fluid curves兲 are contributions from B̃储,
which become potentially significant for Ti = 5Te where␤ ⬃ 1, as well as ion sound wave coupling, which may beneglected only if ␥2 Ⰷ k2储 cs2 for x ⱗ ␳s 关the outer envelope of
the ion response, see Eq. 共B18兲兴. Given k储 = k⬘储 x this condition
may be written as ␥2 Ⰷ k⬘储 2␳s2cs2 or, since ␳s2cs2 = 共12
, ␥2 Ⰷ k⬘储 2v2th共1 + Ti / Te兲␳2e . In the latter form, re+ Ti / Te兲␳2e vth
calling that ␦2 = ␥2 / 共k⬘储 2v2th兲, one sees that this condition isexactly the same as Eq. 共16兲. The proper account of ion
sound wave coupling for Ti ⬃ Te is difficult because ion FLR
effects are not negligible in this case, since ␳s 共the main
scale-length of the ion motion兲 is comparable to ␳i.

Phys. Plasmas 14, 092110 共2007兲

FIG. 5. 共Color online兲 Gyrokinetic 共upper panel兲 and two-fluid 共lower
panel兲 nonlinear reconnection rates for case II vs ␺, the reconnected flux
inside the island, for Ti Ⰶ Te 共solid line兲, Ti = Te 共dotted-dashed兲, Ti = 5Te
共dashed兲.

In the case of the perpendicular ion dynamics, the importance of ion FLR effects has long been appreciated andwas first studied in detail in Ref. 10. This work, like thecalculations presented in Appendix B, is based on a Padé
approximation of the perpendicular kinetic ion response thatis leading-order accurate in the limits of both k⬜␳i Ⰷ 1 andk⬜␳i → 0. In view of the role of ion FLR effects in thesecalculations, it is perhaps surprising that the resulting dispersion relation 关Eq. 共B19兲兴 for arbitrary Ti / Te may also be
obtained from various two-fluid calculations12,16,17,19,20 in thelimit that isothermal electron and ion fluid equations of state
are used. The reason for this agreement, as we show in Appendix B, is that the Padé approximation to the perpendicular
kinetic ion response is exactly reproduced by the usual lowfrequency, reduced-MHD-type approach provided that theion diamagnetic drift is properly included. The numericalsolution to the four coupled reduced two-fluid equations discussed in Ref. 12, shown in Figs. 1 and 2 as the upper
共Ti / Te = 5兲 and lower 共Ti = 0兲 solid lines, go beyond Eqs. 共10兲and 共11兲 to include in the zero-FLR limit ion-sound-wave
coupling and finite ␤ effects. While both of these effects arestabilizing, the fluid predictions for Ti = 5Te are still substantially larger than the GS2 values, indicating the kinetic effects
discussed earlier are important.
IV. NONLINEAR REGIME

Turning now to the nonlinear evolution, we again consider the parameters of case II discussed in Sec. III. As noted
earlier the nonlinear runs have a box size Ly = 2.5Lx, corresponding to ky min␳se = 0.08. This minimum ky corresponds
roughly to the fastest growing linear mode. Following theinitialization of the run with small amplitude random noise,
this longest-wavelenth mode dominates the system as thenonlinear phase is reached. Because this mode falls within in
the large ⌬⬘ regime, the simulations discussed here, aside
from the guide field, are similar to those of the GEM Reconnection Challenge Project.5
Figure 5 shows the nonlinear reconnection rates d␺ / dt

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092110-6

Phys. Plasmas 14, 092110 共2007兲

Rogers et al.

for case II as a function of ␺, the reconnected flux inside themagnetic island. The upper panel shows the GS2 values forTi Ⰶ Te 共solid line兲, Ti = Te 共dotted-dashed兲, Ti = 5Te 共dashed兲.
The nonmonotonic feature in the curve for Ti / Te Ⰶ 1 near␺ ⯝ 0.5 is due to the initial noise in the system, which at
early times leads to competition between the dominant modewith ky␳se = 0.08 and the next-order harmonic. Here ␺ is normalized to B0,maxdi and t to ␥A. At the final times shown in
the figure the island widths are roughly half the box size w⬃ Lx / 2. In all three cases the reconnection continues until theislands span the entire box 共w ⬃ Lx兲, although the reconnection rates drop sharply at times just beyond those shown in
the figure. The lower panel shows the analogous results obtained from a simple isothermal two-fluid simulation withthe same parameters and a large guide field 共Bz ⯝ 10Bx,max兲.
The generalized Ohm’s law in this model includes the Hallterms and electron inertia.5 The reconnection rates in the
gyrokinetic system are comparable to those in the fluid
model and, like the linear gyrokinetic growth rates, exhibit
little dependence on Ti / Te. Numerical tests also show thecontribution of B̃储 in the gyrokinetic equations 共see Appendix
A兲 to both the linear and nonlinear reconnection rates issmall 共ⱗ5 % 兲, in contrast to what one might expect for thehigher Ti cases in which ␤ ⬃ ␤i ⬃ 1.
The fluid code used here was benchmarked in the GEM
reconnection study5 against full-particle simulations and wasfound to produce similar rates of reconnection. Subsequent
work with this same fluid code was carried out, e.g. in Ref.
21, that explored the dependence of the reconnection rate on
the guide field strength, the plasma ␤, and other parameters.
In the strong guide field regime with ␳s ⬎ de relevant to oursimulations, the reconnection rates were characterized by inflow velocities that scale roughly as a small fraction
共⬃0.05兲 of the Alfvén speed based on the reconnecting component of the magnetic field just upstream of the reconnection region,␦Vin ⬃ Vout ⬃ 0.05Vout,l

Vout ⬃

By,up冑4␲nmi .

共17兲

Similar rates have also been observed22 in particle simulations of strong guide field reconnection with a realistic mass
ratio, mi / me = 1836.
In the present system, the reconnecting component of themagnetic field near the reconnection layer is given by
By0共x兲 = By0 max sin共kxx兲 ⯝ B⬘y0x + ¯ ,By,up ⯝ B⬘y0␰,␺ ⯝ 21 B⬘y0␰2 .

共20兲

From the expression for ␺ one finds
1 d␺ 2 d␰=.
␺ dt ␰ dt

共21兲

An expression for d␰ / dt may be obtained from the continuity
equation asd␰␦= Vin = Vout,dtl

Vout ⬃

By,up冑4␲nmi ,By,up = B⬘y0␰

共22兲

or equivalently,d␰ ␦ B⬘y0=␰.
dt l 冑4␲nmi

共23兲

Equation 共21兲 can thus be written as2␦
1 d␺ 2 d␰ 2␦ B⬘y0==
= kxVAy .
␺ dt ␰ dtl 冑4␲nmil

共24兲

Estimating ␦ / l ⬃ 0.05 and normalizing the time units to ␥A as
in Fig. 5, this expression reduces tod␺ 2␦
= ␥A␺ ⯝ 0.1␺ .
dt
l

共25兲

One would therefore expect the slopes of the curves in Fig. 5
at early times to be constant and roughly equal to 0.1, which
indeed they are. The constancy of the slope implies that thereconnected flux and magnetic island width grow exponentially with time.
Concerning the insensitivity of the gyrokinetic system to
Ti / Te, a caveat must be added regarding the level of noisethat is initially imposed on the simulations. The solid curvein Fig. 6 shows the island width as a function of time in the
Ti = 5Te simulation shown in Fig. 5 共top panel兲. The initialnoise level in this simulation is small but finite, such thatvalues of w smaller than about w ⬃ 0.01␳se 共the starting valueof the solid curve兲 cannot be distinguished from the initialnoise. The island width in this case does not saturate andreconnection continues until essentially all the flux in theB⬘y0 = kxBy0 max
共18兲

or in terms of the flux function 共By = ⳵␺ / ⳵x兲:␺ ⯝ 21 B⬘y0x2 .

共19兲

As reconnection proceeds in the nonlinear phase, the flux
surfaces upstream of the reconnection layer at x = 0 are
shifted toward the layer, causing an increase in the local
values of By and ␺. Denoting this time dependent shift in thex direction by ␰共t兲, the values of By and ␺ just upstream ofthe layer at x = 0 can be estimated by replacing x → ␰ in theequilibrium profiles,

FIG. 6. Island widths w / ␳se vs ␥At for Ti / Te = 5 with 共solid兲 and without
共dashed兲 added initial noise in the simulation. The nonlinear saturation is
avoided for very weak levels of initial noise.

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092110-7

Phys. Plasmas 14, 092110 共2007兲

Gyrokinetic simulations of collisionless magnetic reconnection

system is reconnected. When the noise level is drasticallyreduced 共dashed curve兲, however, the growth of the island
halts at about w ⬃ 2␳se ⬃ ␳i. Apparently the saturation mechanism, at least for these parameters, is fragile and easily destroyed by the presence of noise in the system—a feature that
may explain why such saturation behavior has not been previously observed in particle simulations of similar largeguide-field systems 共see, e.g., Ref. 23兲. Such early nonlinear
saturation is not observed in the fluid simulation of the same
system, or in the gyrokinetic simulations at smaller Ti / Te.
Saturation behavior superficially similar to that observed
here has been reported in one particle simulation study24 ofreconnection in a high-beta system without a guide field. It
was conjectured the saturation was due to the formation ofelectron temperature anisotropies 共Te⬜ ⬍ Te储兲, which are
known to have a potentially strong stabilizing impact on thereconnection. In our system the initial plasma equilibrium isisotropic and in the strong guide-field limit the perturbationsof the plasma temperatures, densities, and pressures are small
共see, e.g., Ref. 25 for discussion兲. On the theoretical side, thesaturation of tearing modes has been studied in both low andhigh ␤ plasmas 共e.g., Refs. 26–28兲. Consistent with our findings, in Ref. 28 nonlinear saturation in the strong guide fieldregime is predicted when w ⬃ ␳i due to the interaction of theions with the electrostatic field of the reconnecting mode 共ion
trapping兲. This study, however, considers systems in which
an ambient pressure gradient 共absent in the system considered here兲 leads to a real-frequency of the mode 共also absent
here兲 that is larger than the linear growth rate. Further studyof this issue is needed, therefore, to determine if the physicalmechanism described in that work is consistent with oursimulations.

V. CONCLUSION

We have explored linear and nonlinear collisionless reconnection in the ␳s ⬎ de regime using gyrokinetic simulations. In the linear-theory limit, the gyrokinetic growth rates
are compared to analytic calculations across a range of parameters, including small and large ⌬⬘ and Ti / Te. For thecases in which the analytic calculations of the growth rate aretheoretically reliable for our parameters 共low ␤ and small
Ti / Te兲, reasonably good agreement between the theory andsimulations is found, while for some other cases 共␤ ⬃ 1 andTi ⲏ Te兲, further analytic work is needed to provide a soundbenchmark for the simulations. In particular, the insensitivityof the linear growth rates in the gyrokinetic model to Ti / Teand the plasma ␤, apparently due to the role played by FLR
effects in the parameter regimes considered here, are not well
reproduced by existing analytic calculations. In the nonlinear
regime, the reconnection rates at large ⌬⬘ were explored forvarious values of Ti / Te, and found to be similar to thoseobtained from a two-fluid model. As in the linear case, thedependence of the nonlinear reconnection rates on Ti / Te isquite weak. An exception to this occurs at larger Ti / Te
共Ti / Te = 5兲 and very weak initial perturbations, in which thegrowth of the island width in the gyrokinetic systemsaturates at the ␳i-scale. This saturation for our parameters
may be overcome by adding small levels of initial noise to
the simulations.
ACKNOWLEDGMENTS

This material is based upon work supported by NSF
Grant No. 0238694, the Center for Multiscale Plasma Dynamics, and DOE Grant No. DE-FC02-04ER54784. P.R. is
supported by a EURATOM Fusion fellowship.
APPENDIX A: THE GS2 GYROKINETIC EQUATIONS

The gyrokinetic equations, with species label ␣, may be
written as3

冉

冊

⳵
c
+ i⍀d␣ + v储ⵜ储 h␣共k⬜兲 +⳵t4 ␲ 2B⬙ ⫻ k⬜
⬘ 兲兴 · ␹共k⬜
⬘ 兲h␣共k⬜
⬙ 兲␦k⬘ +k⬙ ,k⬜
⫻ 兺 关b · 共k⬜
⬜

k⬜
⬘ ,k⬜
⬙

=−

冉

+

⬜

冊

⳵ F 0␣
⳵+ i⍀*␣ ␹共k⬜兲q␣
⳵⑀
⳵t

1
2␲

冕

␲

−␲

exp共− iL兲C共f ␣兲d␰ ,共A1兲

where L = 共v ⫻ b · k⬜兲 / ⍀c␣, ␰ is the gyroangle, ⑀ = m␣v2 / 2 is2
/ 2B is the magnetic moment, ␹the particle energy, ␮ = m␣v⬜
is the generalized gyrokinetic potential

␹ = J0冉 冊冉
k ⬜v ⬜
⍀ c␣

␾−

冊 冉 冊

k ⬜v ⬜ v ⬜
v储
A储 + J1␦B储 ,c⍀ c␣ k ⬜c

共A2兲␾ is the electrostatic potential, A储 is the parallel componentof the vector potential, ␦B储 is the perturbation of the parallel
magnetic field, b is the magnetic-field unit vector, ␦k⬙,k⬘ = 0
for k⬙ ⫽ k⬘ and ␦k⬙,k⬘ = 1 for k⬙ = k⬘, ⍀c␣ = q␣B␪ / 共m␣c兲, andC共f ␣兲 is a collision operator 共not relevant to the present
work兲. The nonadiabatic part of the distribution functionh␣ = h␣共⑀ , ␮ , t兲 is related to the total perturbed distributionfunction f ␣ = f ␣共v , t兲 through f ␣ = q␣␾⳵F0␣ / ⳵⑀ + exp共iL兲h␣, or
具f ␣典␰ = q␣␾⳵F0␣ / ⳵⑀ + J0h␣. The frequencies ⍀d␣ and ⍀*␣ aredefined as⍀ *␣ =b ⫻ k⬜ · ⵜF0␣,
m ␣⍀ c␣⳵ F 0␣/ ⳵ ⑀⍀ d␣ = k ⬜ · b ⫻共A3兲m␣v2储 b · ⵜb + ␮ ⵜ B.
m ␣⍀ c␣

共A4兲

The ion and electron “background” distribution functions
F0␣ are assumed to be isotropic Maxwellians. The gyrokinetic equations are coupled to the Maxwell equations, which
may be written as

兺␣

冕 冋

2
A储 = −
ⵜ⬜

4␲
兺
c ␣

册

⳵F0
+ J0h␣ = 0,
⳵⑀

共A5兲

冕

共A6兲

d 3v q ␣ q ␣␾

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092110-8

Phys. Plasmas 14, 092110 共2007兲

Rogers et al.

␦B储 = −

4␲
兺
c ␣

冕

d 3v

q ␣v ⬜
J 1h ␣ .
k⬜

共A7兲

⌫0 ⯝

1
2
1 − ␳2i ⵜ⬜

共B9兲

,

Eq. 共B8兲 simply becomes
APPENDIX B: LINEAR THEORY2␾ = 共1 − ␳2i ⵜ⬜
兲␾i .

Assuming ␥ Ⰷ k储cs, the ion density perturbation n˜i isgiven by 共see, e.g., Ref. 10兲
ñi =

n 0e
关⌫0共b兲 − 1兴␾,Ti

⌫0共b兲 = I0共b兲e−b,2b = − ␳2i ⵜ⬜
,

共B1兲where ␳i = 冑Ti / mi / ⍀ci is the ion Larmor radius and I0 is themodified Bessel function. The perturbed electron continuity
equation gives
c2k⬘储 21A ⬙,ñe ⯝ ⵜ储J̃储 = x
4␲e␥2
␥e

␥
Az ,
A=
ik⬘储 c

共B2兲

where in the linear regime we have assumedⵜ储 = ik⬘储 x,

k⬘储 =

冋 册

ky dBy0k y␥A=,
Bz0 dx x=0 VAJ̃储 ⯝ −

c
A⬙ .
4␲ z
共B3兲

The quasineutrality condition n˜i = n˜e can thus be written as
2 2␥A
k2y ␳seTe关⌫0 − 1兴␾ = x
A⬙ .
␥2
Ti

共B4兲

Finally, the generalized Ohm’s law, assuming an isothermal
electron response, is−m e␥
J̃储 = − ⵜ储共Teñe兲 − n0eE储
e

共B5兲or equivalently, using Eqs. 共B2兲 and 共B3兲−

k 2␳ 2 ␥ 2E储= x␾ + A = y se2 A 共␦2e + x2兲A⬙ ,
ik储
␥

␦2e =

␥
2 ,
k⬘储 2vthek 2␥ 2k 2␥ 2␾i⬙ = x y 2A A⬙ = y 2A ⳵x关x2⳵x共A/x兲兴,␥␥x␾i + A =k2y ␳s2␥A2 2 2
共␦ + x 兲A⬙,
␥2

Te
.
me

␾i ⯝ ␾⬁ = const,Te
2 2关⌫0 − 1兴␾ ⬅ ␳seⵜ ⬜␾ iTi

共B7兲

1 + Ti/Te

.

共B12兲

A ⯝ − ␾⬁共x + 2/⌬⬘兲.

共B13兲− ␾i⬘共0兲 =冉

冊

k2y ␥A2 2␾⬁+ A共0兲 .
␥2 ⌬⬘

共B14兲

The constants in this expression may be related to each otherfrom the solution to Eqs. 共B11兲 and 共B12兲. Assuming␳s ⬎ de, this solution, discussed for example in Ref. 20, turns
out to have two scales: an inner scale ␦ that determines thewidth of the current profile 共i.e., A⬙兲, and an outer scale ␳sthat determines the behavior of ␾i. On the inner scale, the ␾iterm in Eq. 共B12兲 can be neglected and one can take A = A0
+ A1 + ¯ , where A0 = A共0兲 is a constant 共that is, the constant␺ approximation is valid兲, yieldingA0,
共 ␦ + x 2兲
2

⌫=

␥
.
k y ␳ s␥ A

共B15兲Integrating this expression and comparing it to Eq. 共B13兲 forx Ⰷ ␦ one obtains
A0␦

tan−1共x/␦兲 = − ␾⬁2
tan−1共x/␦兲.
␲

共B16兲

In the outer region x Ⰷ ␦, one can neglect the ␦2 term in Eq.
共B12兲 and eliminate A⬙ with Eq. 共B11兲, yieldingx␾i + A = x␳s2␾i⬙ .

共B17兲

Differentiating this equation with respect to x, usingA⬘ ⯝ −␾⬁, and noting ␾i共0兲 = 0, one finds the solution

or equivalently,2␳2i ⵜ⬜
␾i .
关⌫0 − 1兴

␦2eWith this boundary condition, Eq. 共B11兲 may be integrated
from x = 0 to x = ⬁ to yield the constraintA⬘ ⯝ ⌫2

To make progress, we introduce the auxiliary variable ␾idefined as

␦2 =

共B11兲

Note that the contribution of the ␳2i ⳵2x terms arising from Eq.
共B10兲 has simply led to the replacements ␦e → ␦, ␳se → ␳s in
Ohm’s law, or equivalently, Te → Te + Ti. The solutions for ␾iand A for x → + ⬁ must be matched to the asymptotic forms

2

2
=
vthe

␾=

Eliminating ␾ in Eqs. 共B4兲 and 共B6兲 with Eqs. 共B7兲 and共B10兲, respectively, one obtains

A⬙ = A1⬙ ⯝ ⌫2
共B6兲

共B10兲

共B8兲

We demonstrate below that c␾i / B is the ion velocity stream
function in two-fluid theory. Now making the well-known
Padé approximation

␾i ⯝ ␾⬁共1 − e−x/␳s兲.

共B18兲

Finally, returning to Eq. 共B14兲 with this solution, and using
关from Eq. 共B16兲兴 A共0兲 = −2␦␾⬁ / 共⌫2␲兲 = −2de␾⬁ / 共⌫␲兲, one
obtains the dispersion relation

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092110-9⌫3 =Phys. Plasmas 14, 092110 共2007兲

Gyrokinetic simulations of collisionless magnetic reconnection

2 de␥3
3 3 3 =␳s
␲k y ␳s ␥A

冉

1−

冊

⌫␲
.
d e⌬ ⬘vជ i⬜ ⯝

共B19兲

At ⌬⬘ → ⬁ this reduces to Eq. 共10兲 while at small ⌬⬘, neglecting the left-hand side, it reduces to Eq. 共11兲.

k 2␳ 2 ␥ 2
E储 = − ik⬘储 共x␾ + A兲 = − ik⬘储 y se2 A 共␦2e + x2兲A⬙ .
␥

共B20兲

In the inner zone x Ⰶ ␳s, the approximate solution for A⬙ isgiven by Eq. 共B15兲,

冉 冊
␦ +x

A0
= E储共0兲
1 + Ti/Te

冢

冣

Te 2
x
Te + Ti.
␦2 + x2

␦2 +⯝ ik⬘储 x冉冢

␦2e + x2
x

␦2 +

2

冊

冣

or
ñ =

n 0c 2ⵜ ⌽.
B⍀ci ⬜

2⌽ or
Substituting this into Eq. 共B24兲 yields ⌽ = ␾ + ␳2i ⵜ⬜
equivalently,2␾ = ⌽ − ␳2i ⵜ⬜
⌽.

3. Electron drift kinetic treatment

The electrons satisfy the drift kinetic equation,d ⳵= + v储ⵜ储 ,
dt ⳵t
共B29兲

共B22兲

2. Ion velocity stream function ␾i

Here we show the variable ␾i as defined by Eq. 共B10兲 is
proportional to the ion velocity stream function in two fluid
theory. With this interpretation Eq. 共B11兲 becomes, both
physically and mathematically, the ion vorticity equation of
the fluid model. Since Eqs. 共B10兲 and 共B11兲 were con2
Ⰷ 1 asstructed to be accurate 共to leading order兲 for ␳2i ⵜ⬜
2 2well as ␳i ⵜ⬜ → 0, this accounts for why the two-fluid modelyields a correct representation of ion FLR effects in theformer limit.
ជ ⫻, asOperating on the ion momentum equation with B
suming an isothermal ion equation of state, and considering
linear perturbations yields
共B23兲

where f = f 0 + f̃, E储 = −ⵜ储␾ − ⳵Az / ⳵t, and the second termarises from the polarization drift,dc
f 0 ⵜ2 ␾ .
B⍀ce dt ⬜

共B30兲

Defining ␳e = vthe / ⍀ce, vthe = 冑Te / me, ⍀ce = eB / 共mec兲, assumingn0
2
2
2
3 exp关− 共v⬜ + v 储 兲/共2vthe兲兴
共2␲兲3/2vthe

f0 ⯝

共B31兲

and taking for linear perturbations ⵜ储 = ik⬘储 x, ⳵ / ⳵t = ␥, Eq.
共B29兲 yieldsf̃ = −

ef 0
cf 0 2v储
E储 −ⵜ ␾.
Te ␥ + ik⬘储 xv储
B⍀ce ⬜

共B32兲

With this expression, the electron density may be written as
n˜e =

冕

d3v f̃ = − 共1 − I兲

en0 2 2en0 E储−␳ ⵜ ␾,
Te ik⬘储 x Te e

共B33兲

where

where
Ti
ñ.
⌽=␾+
n 0e

共B28兲

Comparing this to Eq. 共B10兲 one sees that ⌽ = ␾i as claimed.

ⵜ⬜ · vជ p f ⯝1 ⳵cẑ ⫻ vជ i⬜ ,vជ i⬜ ⯝ ẑ ⫻ ⵜ⬜⌽ +B⍀ci ⳵t

共B27兲

d 2
d
c
e ⳵f0
f̃ +
f 0 ⵜ⬜
␾=E储,dt
B⍀ce dtme ⳵v储

2
␳se
␾i⬙

Te 2
x
Te + Ti␾⬁e−x/␳s .
x2

共B26兲

共B21兲

In the outer zone x ⬃ ␳s, one may use Eq. 共B11兲 to express A⬙
in terms of ␾i⬙, and then use the expression for ␾i given by
Eq. 共B18兲,E储 = − ik⬘储 x

共B25兲

⳵
n 0c ⳵ 2ⵜ ⌽ñ ⯝ n0ⵜ⬜ · vជ i⬜ =⳵tB⍀ci ⳵t ⬜

From Eq. 共B6兲, the parallel electric field is given byE储 ⯝ − ik⬘储冊

Substituting this into the linearized ion continuity equationand neglecting the parallel ion motion yields

1. Parallel electric field E¸␦2e + x2
2
2

冉

1 ⳵
c
ⵜ ⬜⌽ .
ẑ ⫻ ⵜ⬜⌽ −B⍀ci ⳵t

共B24兲

Here ⌽, aside from a constant factor of c / B, is the ion velocity stream function and with Eq. 共B10兲 we will show in a
moment that ⌽ = ␾i. For ⳵ / ⳵t Ⰶ ⍀ci the second term on theright-hand side of Eq. 共B23兲 is small compared to the first,
leading to the iterative solution

I==冑2␲vthe

冕

冑 冕

e−z
,
1 + ␣ 2z 2

1

2

␲

⬁

0

2

⬁

2

dz

2

e−v储 /共2vthe兲
dv储
1 + ik⬘储 xv储/␥−⬁␣ = 冑2

x

␦e

.

共B34兲

Noting that I ⯝ 冑␲ / ␣ for ␣ → ⬁ and I ⯝ 1 − ␣2 / 2 for ␣ → 0,
one is led to the Padé approximation for 1 − I,

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092110-10

Phys. Plasmas 14, 092110 共2007兲

Rogers et al.
4x2
1−I⯝ 2x + ␦2e

共B35兲

with which Eq. 共B33兲 can be written as

冉

冊

x2 E储
en0
+ ␳2e ⵜ2␾ .
n˜e = −
Te x2 + ␦2e ik⬘储 x

共B36兲

If one neglects the ␳2e term on the right-hand side, this equation becomes identical to Ohm’s law, Eq. 共B5兲, if one expresses J˜储 on the left-hand side of Eq. 共B5兲 in terms of n˜eusing Eq. 共B2兲. Continuing, setting n˜ = n˜ with n˜ given by

Eq. 共B1兲 one finds

冉

i

e

冊

x2
Te
关⌫0 − 1兴␾ + ␳2e ␾⬙
2 共x␾ + A兲 =2
Ti
x + ␦ei

共B37兲

or using Eq. 共B9兲 to eliminate ␾ and rearranging terms,
x共x␾ + A兲 = 共x2 + ␦2兲␳s2␾i⬙ − ␳2e ␳2i 共x2 + ␦2e 兲␾i⵳ .

共B38兲

If one neglects the electron polarization drift term ⬀␳2e on theright-hand side and uses Eq. 共B11兲, this equation reduces to
Eq. 共B12兲. Estimating ␾i⵳ ⬃ ␾i⬙ / ␦2 共valid if the ␳2e term issmall兲 one sees that the ratio of the ␾i⵳ term to the ␾i⬙ termin Eq. 共B38兲 for x ⬃ ␦ is 共Ti / Te兲␳2e / ␦2. The same conclusion
follows from Eq. 共B37兲, neglecting ⌫0 under the assumption␦ Ⰶ ␳ i.
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